In the previous post Sequential spaces, IV, we show that the uncountable product of sequential spaces is not sequential (e.g. the product is not sequential). What is more remarkable is that the product of two sequential spaces needs not be sequential. We present an example of a first countable space and a Frechet space whose product is not a k-space (thus not sequential). For the previous discussion on this blog on sequential spaces and k-spaces, see the links at the end of this post.
Let be the real line and let
be the set of all positive integers. Let
be the space
with the topology inherited from the usual topology on the real line. Let
with the positive integers identified as one point (call this point
). We claim that
is not a k-space and thus not a sequential space. To this end, we define a non-closed
such that
is closed in
for all compact
.
Let where for each
, the set
is defined by the following:
where .
Clearly is not closed as
. In fact in the product space
, the point
is the only limit point of the set
. Another observation is that for each
,
is not a limit point of
. Furthermore, if
for each
where
is an infinite subset of
, then
is not a limit point of
. It follows that no infinite subset of
is compact. Consequently,
is finite for each compact
. Thus
is not a k-space. To see that
is not sequential directly, observe that
is sequentially closed.
Previous posts on sequential spaces and k-spaces
Is there an example of a sequential space whose product with itself is not sequential?
Yes, “the square of a sequential space need not be sequential” (see note in Franklin, “Spaces in which sequences suffice”, 1965, Fundamenta Mathematicae, 57(1), p. 112).